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On Design of Anisotropy Distributions, Applying Lamina Formulas for 2d ResultVisualization

Pedersen, Pauli; Pedersen, Niels Leergaard

Published in:Proceedings of the 10th International Conference on Composite Science and Technology

Publication date:2015

Document VersionPublisher's PDF, also known as Version of record

Link back to DTU Orbit

Citation (APA):Pedersen, P., & Pedersen, N. L. (2015). On Design of Anisotropy Distributions, Applying Lamina Formulas for 2dResult Visualization. In A. L. Araújo, J. R. Correia, & C. M. Mota Soares (Eds.), Proceedings of the 10thInternational Conference on Composite Science and Technology IDMEC.

10th International Conference on Composite Science and Technology

ICCST/10

A.L. Araúo, J.R. Correia, C.M. Mota Soares, et al. (Editors)

c© IDMEC 2015

ON DESIGN OF ANISOTROPY DISTRIBUTIONS,APPLYING LAMINA FORMULAS FOR 2D RESULT

VISUALIZATION

Pauli Pedersen and Niels L. Pedersen

Department of Mechanical Engineering, Solid Mechanics

Technical University of Denmark

pauli@mek.dtu.dk, nlp@mek.dtu.dk

Keywords: Optimal design, Anisotropy, Lamina formula, Result visualization.

Summary: In rotational transformation of constitutive matrices, some practi-cal quantities are often termed invariants, but the invariance relates to an un-changed reference direction. Rotating this reference direction, the practical quan-tities do change and this point is clarified with derived rotational transformationfor these practical quantities. The research background for optimal anisotropicconstitutive matrices is shortly presented. Then design results are applied in a 2Dvisualization of optimized constitutive matrices, that are distributed in a finite el-ement (FE) model where each element has a specific reference direction. Thevisualized distributions of physical quantities are; stiffest material direction, ma-terial stiffest longitudinal constitutive component, level of anisotropy, absoluteor relative shear stiffness and orthotropy test.

1. INTRODUCTION

In free material optimization (FMO), the components of the constitutive ma-trices are optimized and they change in the space of a finite element (FE) model,i.e., they are distributed. The constraints for the non-dimensional description ofthese matrices are; symmetry, positive definite and normalized to unit trace. Theoptimized constitutive matrices should be visualized, butthis is not an easy taskand different techniques are applied in the literature. From the authors point-of-view the visualization should be related to the most important physical quantities,and for 2D problems the traditional lamina analysis is foundvaluable.

Analysis and optimization may be performed without rotational transforma-tions in a common coordinate system with thex-direction as reference. How-ever, the visualizations of the optimized results involve rotational transformation

Pauli Pedersen, Niels L. Pedersen

of material behavior, i.e., of the constitutive matrices. For each element in aFE model, the direction of stiffest material direction is taken as reference direc-tion with stiffest direction defined as the direction of largest longitudinal com-ponents in an optimal constitutive matrix, here termed(α1111)θ with θ being theangle counter-clockwise from the commonx-direction to a direction termed theθ-direction.

The traditional lamina formulas are well suited for localizing θ for a specificelement. Withθ, (α1111)θ determined for all elements the available further phys-ical information is calculated, applying practical parameters(α2, α3, α6, α7)θ asevaluated for elemente in the specific reference directionθe. In the present notethe non-dimensional, normalized practical quantities aregiven notationα, as al-ternative to the often preferred notationQ for corresponding dimensional quan-tities. The note shows that the name invariant is not a god choice. The practicalparameters depend on the reference direction and the relations to the commonx-direction are derived.

Although written in relation to 2D constitutive matrices, the approach is alsovalid for 2D structural stiffness matrices[S], 2D structural flexibility matrices[F ], and 2D strength matrices in stress space[H] or in strain space[G]. Alsolaminate stiffness sub-matrices and laminate flexibility sub-matrices may be an-alyzed similarly.

The main readers in mind are researchers with interest in laminate formula-tion, but it is found necessary to give a short introduction in Section 2. to optimalconstitutive matrices, before the application of laminateformulation is detailedin Section 3. Especially the discussion on "invariant" parameters should be noted.Finally, fields for constitutive matrices are exemplified with a suggested visual-ization for the optimal constitutive design obtained in Pedersen and Pedersen(2015).

2. OPTIMAL DESIGN OF CONSTITUTIVE MATRICES

In recent research simple formula for design of constitutive matrices are ob-tained, related to different static as well as to eigenfrequency optimal designproblems. It is shown that for quite different design objectives, the elastic energydensity plays a major role and the results are expressed directly by the currentstrains, with unit matrix norms and separated from the amount of material.

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Pauli Pedersen, Niels L. Pedersen

2.1 Separation from the amount of material

The distribution of material in a continuum is separated into two steps: firstlyhow much material to be used in a reference volumeVe? and secondly how thislocal material should be used to obtain an optimal local constitutive matrix? Bythis separation a clear measure of the total amount of material is possible.

The total amount of material volumeV is constrained and this constraint isassumed to be active, i.e., all material is assumed to be used. This assumption isessential for the obtained optimality criteria. Withρe as local, non-dimensionaldesign parameters for density, this is written

∑

e

ρeVe = V with size limits

0 < ρmin ≤ ρe ≤ ρmax ≤ 1 and the major constraint is written

g =∑

e

ρeVe − V = 0 ⇒ ∂g/∂ρe = Ve (1)

In Pedersen and Pedersen (2015) the theory and procedures for iterative opti-mization to obtain the densitiesρe are presented and will not be further com-mented in the present paper.

The separated local (elemente) constitutive matrix[Ce] is

[Ce] = ρeE0[Ce] = ρeE0

(C1111)e (C1122)e

√2(C1112)e

(C1122)e (C2222)e

√2(C2212)e√

2(C1112)e

√2(C2212)e 2(C1212)e

(2)

whereE0 is a fixed value of modulus,ρe a current local, non-dimensional densityand[Ce] is a non-dimensional matrix, normalized to unit trace as well as to unitFrobenius norm. The discussion of this matrix is of primary interest.

2.2 Constraint for the non-dimensional constitutive components

The constitutive matrices are constrained to be symmetric and positive semi-definite and furthermore normalized such that the FrobeniusnormFe = F ([Ce])is equal to 1 for all elements, here stated in terms of the squared normF 2

e

he = F 2

e − 1 = 0 (3)

With a design objectiveΦ and only the constraint (3), the necessary conditionfor optimality is proportionality between the gradients ofthe objective and the

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Pauli Pedersen, Niels L. Pedersen

gradients of the constraint

∂Φ

∂(Cijkl)e

= λ∂he

∂(Cijkl)e

(4)

where for 2D problems(Cijkl)e is one of the six independent components of theconstitutive matrix andλ is a common factor for all six of these components,related to a specific constitutive matrix.

With F 2

e defined as the sum of the squared components of the matrix[Ce] in(2)

F 2

e = (C2

1111)e + (C2

2222)e + 4(C2

1212)e + 2(C2

1212)e + 4(C2

1112)e + 4(C2

2212)e

(5)

the gradients of the constrainthe = F 2

e − 1 = 0 are directly

∂he

∂(C1111)e

= 2(C1111)e,∂he

∂(C2222)e

= 2(C2222)e,∂he

∂(C1212)e

= 8(C1212)e,

∂he

∂(C1122)e

= 4(C1122)e,∂he

∂(C1112)e

= 8(C1112)e,∂he

∂(C2212)e

= 8(C2212)e (6)

The gradients of the objective, i.e. the left hand side of (4)for specific opti-mization objectives are derived below.

2.3 Compliance or total elastic energy as objective

Compliance is, for design independent loads, equal to the total elastic energyU (twice the total strain energy) and a gradient ofU , say with respect to theconstitutive components(Cijkl)e, can be determined in a fixed strain field (fixeddisplacements field)

∂U

∂(Cijkl)e

= −(∂U

∂(Cijkl)e

)fixed strain = −VeρeE0(∂ue

∂(Cijkl)e

)fixed strain (7)

whereVe is the volume in which we have constant strains{ǫ}e and the constantconstitutive matrix[C]e. Expanding the non-dimensional matrix productue =

{ǫ}Te [C]e{ǫ}e with {ǫ}T

e = {ǫ11, ǫ22,√

2ǫ12}e give

u =(C1111)e(ǫ2

11)e + (C2222)e(ǫ

2

22)e + 4(C1212)e(ǫ

2

12)e+

2(C1122)e(ǫ11)e(ǫ22)e + 4(C1112)e(ǫ11)e(ǫ12)e + 4(C2212)e(ǫ22)e(ǫ12)e (8)

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Pauli Pedersen, Niels L. Pedersen

and the gradients are

∂U

∂(C1111)e

= ρeVeE0(ǫ11)e(ǫ11)e,∂U

∂(C2222)e

= ρeVeE0(ǫ22)e(ǫ22)e,

∂U

∂(C1212)e

= 4ρeVeE0(ǫ12)e(ǫ12)e,∂U

∂(C1122)e

= 2ρeVeE0(ǫ11)e(ǫ22)e,

∂U

∂(C1112)e

= 4ρeVeE0(ǫ11)e(ǫ12)e,∂U

∂(C2212)e

= 4ρeVeE0(ǫ22)e(ǫ12)e (9)

2.4 Multiple load cases and resulting optimality criterionfor compliance optimizations

With multiple load cases, all design independent, numberedn = 1, 2, ...N ,the gradients (9) holds for each load case. The corresponding strains(ǫ11)n,(ǫ22)n and(ǫ12)n are all determined in the same coordinate system. Therefore,the simple optimization of minimizing a linear combinationof compliance’s ex-pressed in the energiesUn is

Minimizing U =∑

n

ηnUn for he = F 2

e − 1 = 0 (10)

for given weight factorsηn, say with∑

n ηn = 1.The design for the multiple load case, that satisfies the optimality criterion is

(Cijkl)e = λ∑

n

ηn((ǫij)e(ǫkl)e)n (11)

a simple optimal design result. The case of a single load caseis further simplified

(Cijkl)e = λ(ǫij)e(ǫkl)e (12)

as seen directly by inserting (6) and (9) in (4).

2.5 Gradients and resulting optimality criterionfor single eigenfrequency optimization

The local gradient of the Rayleigh quotient with respect to the components ofthe local constitutive matrix is simple when the mass distribution is unchanged

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Pauli Pedersen, Niels L. Pedersen

(kinetic energiesT andTe unchanged), here with hat notation as an alternativeto extended index of fixed displacements

∂ω2

∂(Cijkl)e

=∂(U/T )

∂(Cijkl)e

=∂(U/T )

∂(Cijkl)e

) =∂(Ue/Te)

∂(Cijkl)e

=1

Te

∂Ue

∂(Cijkl)e

=ρeVeE0

Te

∂ue

∂(Cijkl)e

with fixed strains inue = {ǫ}Te [C]e{ǫ}e (13)

From the final relation in (13) then follows

∂ω2

∂(C1111)e

=ρeVeE0

Te

(ǫ11)e(ǫ11)e,∂ω2

∂(C2222)e

=ρeVeE0

Te

(ǫ22)e(ǫ22)e,

∂ω2

∂(C1212)e

= 4ρeVeE0

Te

(ǫ12)e(ǫ12)e,∂ω2

∂(C1122)e

= 2ρeVeE0

Te

(ǫ11)e(ǫ22)e,

∂ω2

∂(C1112)e

= 4ρeVeE0

Te

(ǫ11)e(ǫ12)e,∂ω2

∂(C2212)e

= 4ρeVeE0

Te

(ǫ22)e(ǫ12)e (14)

that except for a factor is identical to (9).Comparing with (11) and (12) it is seen that the optimality criterion for the

discussed different 2D plane problems is for all of them satisfied for

(Cijkl)e =(ǫijǫkl/(ǫ

2

11+ ǫ2

22+ 2ǫ2

12))

e(15)

now written with the appropriate normalization.

2.6 Proof of unit norms

The result (15) shows that[Ce] = {α}{α}T is described by such a dyadicproduct. Then by definitions of trace and Frobenius norms follows, that thevalues of trace and Frobenius norms are always equal and[Ce] is semi-positivedefinite.

trace[Ce] = Frobenius[Ce] = {α}T{α}where {α}T{α} > 0 for {α} 6= {0} (16)

Omitting the indexe for element we proceed the discussion of the obtainedconstitutive matrix as described directly by the corresponding strain state. Al-though a constitutive matrix is not necessary obtainable asa dyadic product, this

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Pauli Pedersen, Niels L. Pedersen

will be the case for the optimal constitutive matrix, where the important result in2D plane problems with normalization to unit norms is

[C] = {α}{α}T with {α}T = {ǫ11 ǫ22

√2ǫ12}/

√ǫ2

11+ ǫ2

22+ 2ǫ2

12(17)

That the optimal constitutive matrix of unit norms in 2D is described by onlythree parameters (the strain components) limits the possibilities for a matrix withnormally up to 6 independent parameters. An example is that an isotropic[C] isonly possible with zero Poisson’s ratio and for this case{α}T = {1 1 1}/

√3.

Numerically the rate of change of the constitutive matricesare in each re-design limited by a non-dimensional step parameter0 ≤ β ≤ 1 similar to the de-sign approach for strength optimization in Pedersen and Pedersen (2013) whereβ = 0.5 andβ = 0.1 were used, i.e.,

[C]new = β[C]from (17) + (1 − β)[C]old (18)

The design approach is initiated with[C]0 = [I]/3, i.e., zero Poisson’s ratioisotropic material, positive definite, non-dimensional and normalized. It is con-cluded that for a given strain state the optimized non-dimensional constitutiveconstitutive matrix is known with unit trace and Frobenius norm. Note, that withinitial positive definite[C] it will for β < 1 stay positive definite through the re-design iterations. Numerical valueβ = 0.2 is applied for the visualized examplein Section 4., and even with this rather lowβ value fast convergence is obtained.

3. VISUALIZATION OF FIELD OF CONSTITUTIVE MATRICES

Visualization of fields of 3× 3, symmetric, positive definite constitutive ma-trices of unit norms is based on formulations from laminate theory. Practicalparameters that often are termed invariants are valuable, but there seems to be aneed for discussion of the property "invariant".

3.1 Use of laminate formula

For anisotropic material the anisotropy should be visualized, but without go-ing into all details of the six 2D components. A 2D material non-dimensionalconstitutive matrix[C] is given in a global x, y coordinate system with the x-

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Pauli Pedersen, Niels L. Pedersen

direction as the reference direction by

[C] =

α1111 α1122

√2α1112

α1122 α2222

√2α2212√

2α1112

√2α2212 2α1212

x

(19)

with the assumed condition that[C] is positive definite and the trace of the posi-tive diagonal elements is normalized to unity, i.e.,

α1111 + α2222 + 2α1212 = 1 (20)

These conditions then hold in any rotated coordinate system. A physical de-scription of the constitutive matrix is of major interest, so the direction of largestlongitudinal material stiffness must be located.

According to laminate theoryα1111 as a function of rotation, termed(α1111)θ,is given by the six components in thex reference coordinate system, here chosenin a form linear in trigonometric factors,

(α1111)θ =(α1111 + α2222)x/2 + (α2)x cos(2θ) − (α3)x(1 − cos(4θ))+

(α6)x2 sin(2θ) + (α7)x sin(4θ) (21)

where the practical parameters are defined by

(α2)x = (α1111 − α2222)x/2

(α3)x = (α1111 + α2222 − 2(α1122 + 2(α1212))x/8

(α6)x = (α1112 + α2212)x/2

(α7)x = (α1112 − α2212)x/2 (22)

For orthotropic materialsα6 = α7 = 0 in specific directions, but for the freematerial this will not always be the case, so we analyze the more general case.Several extremum solutions for(α1111)θ may exist in the interval0 ≤ θ < π. Tolocate the maximum of(α1111)θ, the function (21) is numerically evaluated at anumber ofθ values (here chosen with increments∆θ = π/1800). This can bedone for each elements andθe is then the angle for the largest value(α1111)θe.The values of(α1111)θ has an upper bound of 1 and a lower bound of 1/3. Thisfollows from the trace being 1, and having positive eigenvalues in this interval.This then also follows for the non-dimensional longitudinal stiffness. For highvalues of(α1111)θ a single fiber direction is approached and for lower values of(α1111)θ an isotropic material with zero Poisson’s ratio material isapproached.

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Pauli Pedersen, Niels L. Pedersen

Similar to (21) the remaining constitutive components withtheθ-direction asreference direction may then be evaluated by

(α2222)θ =(α1111 + α2222)x/2 − (α2)x cos(2θ) − (α3)x(1 − cos(4θ))−(α6)x2 sin(2θ) + (α7)x sin(4θ)

(α1122)θ =(α1122)x + (α3)x(1 − cos(4θ)) − (α7)x sin(4θ)

(α1212)θ =(α1212)x + (α3)x(1 − cos(4θ)) − (α7)x sin(4θ)

(α1112)θ =(α2)x sin(2θ)x/2 − (α3)x sin(4θ)) + (α6)x cos(2θ) + (α7)x cos(4θ)

(α2212)θ =(α2)x sin(2θ)x/2 + (α3)x sin(4θ)) + (α6)x cos(2θ) − (α7)x cos(4θ)(23)

all this well known from laminate theory.

3.2 Discussion on "invariant" parameters

The definitions of(α2)θ, (α3)θ, (α6)θ), (α7)θ with reference to a specificθ-direction are defined by

(α2)θ = (α1111 − α2222)θ/2

(α3)θ = (α1111 + α2222 − 2(α1122 + 2(α1212))θ/8

(α6)θ = (α1112 + α2212)θ/2

(α7)θ = (α1112 − α2212)θ/2 (24)

and their numerical values may be different from the parameters in (22). Thefollowing relations are derived by inserting (21) and (23) in (24)

(α2)θ = (α2)x cos(2θ) + (α6)x2 sin(2θ) (= (α2)x for θ = 0 and π)

(α3)θ = (α3)x cos(4θ) + (α7)x sin(4θ) (= (α3)x for θ = 0 and π)

(α6)θ = (α6)x cos(2θ) − (α2)x sin(2θ)/2 (= (α6)x for θ = 0 and π)

(α7)θ = (α7)x cos(4θ) − (α3)x sin(4θ) (= (α7)x for θ = 0 and π) (25)

Material orthotropy imply zero of the following parameter combinations

zx = (α7)x(α2)2

x − 4(α7)x(α6)2

x − 4(α6)x(α3)x(α2)x

zθ = (α7)θ(α2)2

θ − 4(α7)θ(α6)2

θ − 4(α6)θ(α3)θ(α2)θ (26)

If zx is zero, then the material is orthotropic and alsozθ is zero, because thecondition (26) holds in any coordinate system. The derived functions(25) fulfillsthis, by setting(α6)x = (α7)x = 0.

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Pauli Pedersen, Niels L. Pedersen

The conclusion from the present analysis is that the parameters (24) as wellas (22) should be termed practical parameters instead of invariant parameters.

3.3 Important anisotropy quantities

It is suggested for the constitutive matrices of an optimized design to presentthe following five distributions

• Largest longitudinal stiffness by(α1111)θe for all elementse in a color plot,noting the limits1/3 ≤ (α1111)θe ≤ 1 with 1/3 for isotropy with zeroPoisson’s ratio and with 1 for unidirectional fiber.

• Direction of largest longitudinal stiffnessθe for all elementse by direc-tional lines, noting the limits0 ≤ θe ≤ π. May be combined with thecolor plot above.

• Level of anisotropy by2(α2)θe = (α1111)θe − (α2222)θe for all elementse ina color plot, noting the limits 0 and 1 with 1 for high level of anisotropyand 0 for symmetry.

• Relative importance of shear stiffness by8(α3)θe for all elementse in acolor plot. High shear stiffness corresponds to negative values of8(α3)θe,i.e. 4α1212 > α1111 + α2222 − 2α1122, as seen in (24). Alternatively,α1212

may be directly visualized.

• Test for material orthotropy byze for all elementse in a color plot. Onlyplaces withze = 0 have orthotropic material. The color plot relates to ascaled, squared test quantity with a lower limit to identityorthotropy.

3.4 Other matrices with similar rotational transformations

The present note written in relation to 2D constitutive matrices, is also validfor 2D structural stiffness matrices[S], 2D structural flexibility matrices[F ],and 2D strength matrices in stress space[H] or in strain space[G]. Also lami-nate stiffness sub-matrices and laminate flexibility sub-matrices may be analyzedsimilarly.

4. VISUALIZATION EXAMPLE FROM OPTIMAL ANISOTROPY

In Pedersen and Pedersen (2015) a cantilever (with fixed material at the tip)is optimized to maximize the first eigenfrequency. Without specifying here thedetails of analysis and optimization by iterative redesign, we visualize in Figure1 the obtained constitutive matrices, as suggested above inSection 3.3.

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Pauli Pedersen, Niels L. Pedersen

a) Largest longitudinal stiffness by(α1111)θb) Level of anisotropy by2(α2)θ

c) Relative shear stiffness by8(α3)θ d) Orthotropic material, only ifz2

θ = 0

Figure 1: Visualization of distributions for constitutive matricesadded direction of largest lon-gitudinal stiffness in Figure 1a. The white spots near the tip of fixed material are places ofminimum material density, being the same in all Figures 1a-d. The further white areas in Figure1d contains materials classified as orthotropic.

Note, that the quantities in Figures 1a-c are measured in individual rotatedcoordinate systems that are visualized by the directions inFigure 1a.

For largest longitudinal stiffness in Figure 1a we see blue color (close to 0.85)at the upper and lower boundaries, yellow color (close to 0.45) at the "beam axis"and green color between these zones. All this as expected in relation to the mostsimple bending eigenmode. For direction of largest longitudinal stiffness addedin Figure 1a, the 45 degrees at the "beam axis" and parallel to the upper andlower boundaries also agree with simple bending of a cantilever.

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Pauli Pedersen, Niels L. Pedersen

The distribution of level of anisotropy is visualized by2(α2)θ in Figure 1band show small relative values of(α2222)θ by blue color close to upper and lowerboundaries, i.e., high level of anisotropy. Close to symmetry (α2222)θ ≃ (α1111)θ

by red color close to the "beam axis".The relative importance of shear stiffness is visualized bydistribution of

8(α3)θ in Figure 1c. For high relative shear stiffness this quantity will be nega-tive. For the present case only positive values are found.

The distribution of possible orthotropy is visualized by a scaledz2

θ in Figure1d, where the zero limit is set to 0.001. The white areas (awayfrom the tip) arethus areas of material orthotropy, without showing the directions of orthotropy.

5. Conclusion

Visualization of results from optimal design may not be too complicatedin traditional size, shape or topology design, but in free material optimization(FMO) constitutive matrices in the continuum or structuralspace are part of theobtained design. A visualization of such distribution of matrices for 2D problemswith 6 different matrix components is demonstrated.

From laminate analysis, the formulation for rotational transformation is ap-plied and is found useful. Practical parameters that usual are stated as invariantsare an important part of this formulation, but the notion invariants needs to bediscussed, because it only relates to a specific reference direction. The visual-ized distributions of physical quantities are; stiffest material direction, materialstiffest longitudinal constitutive component, level of anisotropy, absolute or rel-ative shear stiffness and orthotropy test.

Optimal design of material distribution is often effectively obtained by designiterations based on a stated optimality criterion. It is recently found that theoptimal constitutive matrices (the anisotropy) is simply related to the actual strainfield(s). Since this is not well known, it is chosen to shortlydescribe the theorybehind this result as an introductory to the visualization aspects.

References

Pedersen, P., and Pedersen, N. L. (2013). On strength design usingfree material sub-jected to multiple load cases.Struct. Multidisc. Optim. 47(1):7–17.

Pedersen, P., and Pedersen, N. L. (2015). Distributed material densityand anisotropyfor optimized eigenfrequency of 2D continua.Struct. Multidisc. Optim. DOI:10.1007/s00158-014-1196-6

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